estimation of bearing capacity and...

Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online)

An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/02/jls.htm

2015 Vol. 5 (S2), pp. 3038-3050/Mohammadizadeh and Asadi

Research Article

© Copyright 2014 | Centre for Info Bio Technology (CIBTech) 3038

ESTIMATION OF BEARING CAPACITY AND SETTLEMENT OF

SPREAD FOOTINGS OVER STONE-COLUMN-REINFORCED CLAY

USING FUZZY MODELS AND ARTIFICIAL NEURAL NETWORKS

*Mohammadizadeh M and Asadi M

Department of Engineering, Sirjan Branch, Islamic Azad University, Sirjan, Iran *Author for Correspondence

ABSTRACT

Stone columns, also known as granular piles, have been extensively used to improve bearing capacity and

reduce settlement of foundations lying over soft soils. However, due to uncertainties existing in behavior of the column’s material and its surrounding soil, mechanical response of this composite ground is not

fully understood. Recently, some efforts have been made to estimate bearing capacity and settlement of

spread footings on the composite ground more accurately using statistical analyses. Although significant

results have been reported in these studies, the proposed statistical models may have some shortcomings and do not fit well to the experimental data. In current study, fuzzy models and artificial neural networks

have been used to evaluate mechanical behavior of the composite ground. In all cases, an information

criterion is applied to obtain the most optimized architecture of the models. This is particularly recommended when a few number of high quality data is available for training and testing the models. It

is shown, through comparison with experimentally measured data, that the proposed models can

accurately estimate bearing capacity and settlement of foundations on the ground reinforced with stone columns.

Keywords: Stone Column; Bearing Capacity; Clayey Soil; ANFIS; Artificial Neural Networks

INTRODUCTION

Clay soils are of lower strength compared with granular soils such as sand and gravel. This implies that

when a heavy building is constructed over a clayey bed, either the soil is failed in shear or significant settlement is occurred both of which result in serious problems in operation of the structure. One way of

solving this problem is drilling some holes at specified intervals and filling them with granular material

prior to construction of the foundation. These structures are terminologically called stone columns or aggregate piers and the set of soil - stone columns is known as composite or reinforced ground. The

bearing capacity and settlement of the composite ground may not be easily determined since the

mechanical behavior of these two basic elements, i.e. clay and stone columns, is of high complexity and

cannot be adequately explained through theoretical methods such as the theory of elasticity. Moreover, the combination itself and the resulting interactions should be taken into consideration. In such

circumstances, usually the numerical modeling techniques, e.g. finite element method, are regarded as the

first and cheapest method of analysis. The other and more reliable solution is conducting field tests on the real ground and assessing its behavior. However, such tests are expensive and require a wide range of

equipments as well as skilled and experienced manpower. Development of accurate and applied

techniques for estimating the bearing capacity and settlement of clay ground reinforced with stone

columns is the issue addressed in this paper. As one of the earliest attempts for analyzing the behavior of stone columns, Greenwood (Greenwood,

1970) studied the behavior of a single stone column using the theory of plasticity. Afterwards, Brauns

(1978) and Van et al., (1997) have proposed modifications to this initial equation. In addition, Fox and Cowell (1998) and Wissmann (1999) presented some equations similar to the well-known relations used

to evaluate the bearing capacity of shallow foundation (such as Terzaghi’s relations (1943)). On the other

hand, Balaam et al., (1977) and Balaam and Booker (1988) were the first who analyzed the behavior of these columns using simple assumptions of the theory of elasticity. Noorzad et al., (1997) assumed the

behavior to be elasto-plastic and calculated the total settlement of the ground. Guetif et al., (2007)




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investigated the response of the clayey bed reinforced with stone columns using vibro-compaction

technique. The main objective of this study was estimation of the Young's modulus of the composite

ground which was achieved through a finite element analysis using PLAXIS software. This study revealed that construction of stone columns using the aforementioned technique results in initial pore

water pressure in the clay environment which is gradually vanished over the time. Consequently, the

properties of consolidated clay (including deformability and Young’s modulus) are improved. Based on the results of modified triaxial tests on the soil samples reinforced with cemented piles, Juran

and Riccobono (1991) assessed the influence of cementation as well as the grouping effect on the

settlement of the composite ground. After analyzing the relationship between axial strain and stress ratio

and the relationship between axial strain and volumetric strain, Pooroshasb and Meyerhof (1997) derived a correlation between the applied load and the settlement for the stone-columns-reinforced grounds.

The idea of encasing the stone columns has been discussed in recent years (Gniel, 2009). This is

especially important in very soft soils where either the construction of ordinary stone columns is not generally feasible or the columns bugle under the loading effects making them practically useless. Other

solutions have been also proposed for very soft soils, such as putting horizontal geogrid layers inside the

stone columns (Sharma et al., 2004). Nevertheless, such techniques of implementation cause practical problems which may reduce relative merits of stone columns, therefore, these techniques are only

recommended in the cases where the surrounding clay is of very low strength.

During past decade, computational intelligence techniques have been successfully applied for modeling

complex systems in science and engineering. In current study, mechanical behavior (bearing capacity and settlement) of clay bed reinforced with stone columns is evaluated using fuzzy models as well as artificial

neural networks. The data required for training and verifying the proposed models are collected from

published reports in the literature. In addition, the results are compared with recently published researches and their relevance is demonstrated.

Stone Columns

Stone columns also known as granular piles and aggregate piers are one of the techniques used to improve

soft soils and loose strata. Stone columns are cylindrical elements which gain their strength and stiffness from the confinement provided by the surrounding soil. This implies that the lateral (radial) strain

developed under loading and the resulting interaction between soil and column mobilize the strength of

stone-column-improved ground. Due to their efficiency, easy construction and availability of the material with low price, stone columns

are regarded as a very common method of soil improvement. These columns are a vertical support for

their overlying embankments and structures. Thy decrease the settlement and increase the bearing capacity. In fine-grained impermeable soils, they act as vertical drains and hence reduce the time of

consolidation. Moreover, if the soil is cohesion-less, like sands and silts, this drainage function helps to

mitigate liquefaction potential under cyclic loading of earthquake.

Stone columns may be used either in linear configuration to support walls and strip foundations or as triangular/rectangular group to support mat foundations and embankments. In the cases which stone

columns are employed to support a vertical load (of a foundation for example), the bearing capacity of

column itself is of importance. On the other hand, in some occasions such as earth slopes, stone columns rely on their drainage capabilities to increase the stability of the slope.

Stone columns may be constructed through a variety of techniques such as vibro-replacement, vibro-

flotation, pre-drilling and encased borehole methods. As shown in Figure 1, in vibro-replacement method, a long torpedo-shaped probe is driven into the soil either by the vibration only (dry method) or by a

combination of vibration and water jet (wet method). Once the probe excavated the ground up to the

desired depth, the aggregates are poured into the hole and compacted by the vibratory probe. Pre-drilling

method has been used during past decade as a relatively low-cost alternative of other techniques which require expensive equipment and skilled workforce. In this method, the aggregate material are dumped

into a hole, which is previously excavated using auger, and then compacted by falling weights. More

details on construction of stone columns may be found in the literature (Barksdale and Bachus, 1983).




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Figure 1: Construction of stone columns by vibro-replacement technique (after Hayward Baker,

2012)

Analyzing the Behavior of Stone Columns

Bearing Capacity Estimation Brauns (1978) demonstrated that the ultimate bearing capacity of stone column with the surcharge q over

the surrounding soil can be estimated as:

psu

ult

sqq

2tantan

tan1

2sin

2

(1)

Where su is undrained shear strength of the native soil; p and s are failure angle in aggregate material of the column and the surrounding soil respectively; is the conical angle of shearing in the native soil. It

can be shown that for a typical value of p = 45°, the conical angle of friction equals to 62°.

Mitchell (1981) proposed a simple empirical method to estimate the ultimate capacity of a single isolated stone column:

puult Nsq . (2)

where Np is bearing capacity factor recommended to be 25. Barksdale and Bachus (1983) suggested Np to

be between 18 and 22 for stone columns constructed by vibration based on back- analysis of field tests

results. Moreover, Bergado and Lam (1987) recommended Np ranging from 15 to 18 for stone columns constructed using falling weights (rammed method). Stuedlein and Holtz (2013) proposed the following

empirical modification to equation (above):

Np = exp(0.0096. su + 3.5) (3) Barksdale and Bachus (1983) also proposed the following equation to evaluate the bearing capacity of a

group of stone columns:

245tan2

245tan2

3,ave

aveuave

groupult sq

(4)

Where ave and su ave are friction angle and undrained shear strength of the composite ground

respectively. 3 is minor principal stress applied radially around the stone column. Stuedlein and Holtz (2013) performed multiple linear regression (MLR) modeling of the bearing capacity of stone-column-

treated ground and obtained the following equation:

mp

mp

rpfrrpult SdaSq

005.071.13

07.0914.1013.0756.4)ln(

(5)




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Where Srp is the slenderness ratio of the column, i.e. the ratio of its lengths to diameter. It will be noted

that Srp plays a significant role in the value of bearing capacity; ar is area replacement ratio which is

defined as the ratio of column’s cross section to the area of unit cell (see the next section); df is depth of footing embedment; and mp is the native soil shear mass participation factor defined as the ratio of su to

ar.

Settlement Analysis So far several approaches have been proposed to estimate the settlement of foundations resting over stone

columns, e. g. (Fox and Cowell, 1998; Lawton et al., 1994; White et al., 2006). Most of these methods

assume that the unit cell concept, which is described in detail by Barksdale and Bachus (1983), is valid.

This theory states that the behavior of a group of columns beneath a uniformly loaded area can be simplified to a single column installed at the centre of a cylinder of soil representing the column’s zone of

influence.

In order to calculate the settlement of stone column improved ground, Lawton et al., (1994) and Fox and Cowell (1998) proposed the ground to be divided into two separate zones namely the upper and lower

zone. The settlement of the upper zone is estimated assuming that the surrounding soil and the stone

column act as independent elastic springs. On the other hand, settlement of the lower zone is evaluated either with Westergaard stress distributions or by assuming the two-layered soil profile (a stiff, infinite,

uniform elastic upper layer and a lower layer representing the unimproved soil).

A MLR analysis on the measured displacements ranging from two to 50 mm with respect to common

design variables is reported by Stuedlein and Holtz (2014):

rpfmprppr SdbbSbLbabbq 543210)ln( (6) Where bi (i = 0,5) are statistical constants; Lp is the length of stone column. Stuedlein and Holtz (2014)

showed that the MLR model conforms well to the measured values of bearing pressures (q) for a wide

range of displacements and various configurations.

Computational Intelligence Systems

Artificial Neural Networks

From a general point of view, ANNs can be divided into two types namely Feed-forward Neural

Networks (FNN) and Recurrent Neural Networks (RNN). In FNNs the input signal only passes forward while in RNNs neurons send feedback signals to each other. The main advantage of FNNs is their ease of

implementation and estimation of inputs and outputs. However, they slow down the training procedure

and require a large amount of training data. On the other hand, RNNs take advantage of their internal state known as ―memory‖ to rapidly extract the relationship between the data (White, 1996).

Probabilistic Neural Network (PNN) introduced by Specht (1990) is a kind of FNN, which performs

classification tasks where the target variable is categorical. PNN can sustainably increase the learning speed in comparison with conventional algorithms such as

back propagation. The use of a PNN is especially advantageous due to its ability to converge the

underlying function when only few training samples are available. Generalized Regression Neural

Network (GRNN) in particular, is of the same architecture as PNN except that they perform regression where the target variable is continuous.

Adaptive Neuro-Fuzzy Inference System

Fuzzy logic first introduced by Zadeh (1965) is a solution for efficient and flexible modeling of complex systems that cannot be easily modeled using conventional mathematical techniques. The start point of

developing a fuzzy model is derivation of fuzzy if-then rules. Due to their training capabilities, ANNs can

form an appropriate relationship between input and output variables. On the other hand, fuzzy systems are

regarded as ideal tools to deal with nonlinear problems associated with variables of approximate nature. Therefore, combination of fuzzy inference system and artificial neural networks is a brilliant idea to

develop a powerful approach of modeling.

Adaptive Neuro Fuzzy Inference System (ANFIS) is such a method in which the relationship between input/output variables is established by the fuzzy portion, and the parameters of fuzzy membership

functions are optimized by ANN learning algorithm (Jang, 1995).




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Fuzzy Clustering

Fuzzy clustering, which falls into unsupervised learning paradigm, is an automatic procedure during

which the similar data samples are categorized a number of clusters. Similarity may be defined according to various criteria. In distance-based clustering, for instance, the samples which are adjacent to each other

are considered to be similar.

Fuzzy C-Means Clustering Like its classical algorithm, the number of clusters (c) in fuzzy c-means clustering should be specified

before. The objective function is as follows:

c

i

n

k

ik

m

ik

c

i

n

k

ik

m

ik vxuduJ1 1

2

1 1

2

(7) where xk is k-th sample and vi is the representative or the center of i-th cluster. uik shows the degree of

dependence of i-th sample to the k-th cluster. In this equation, m is a real number greater than 1, and usually set as 2 (Stuedlein and Holtz, 2013). The sign ||*|| denotes a function which expresses the degree

of similarity (here the distance) to the cluster center. uik generates the matrix u with c rows and n columns

and the elements taking a value between 0 and 1. If all of the elements take either 0 or 1 the algorithm

turns into the classical type. Fuzzy Subtractive Clustering

The main advantage of fuzzy subtractive clustering is preventing from unnecessary and excessive growth

of cluster numbers. In this algorithm, each data point (xi) is regarded as a potential cluster center for which the density measure is calculated using the following equation:

n

j a

ji

ir

xxD

12

2

)2/(exp

(8)

where ra is a constant greater than zero called radius of influence. According to this equation, the point

with more samples in their vicinity will have a greater density measure. Once the density measure is calculated for all of the samples, the point with the highest value of density measure is selected as the first

cluster center. Subsequently, the density measure for other points is modified using the equation below:

2

2

1

1)2/(

expb

ci

ciir

xxDDD

(9) where rb is a constant which is recommended to take a value equal to 1.5 (Stuedlein, 2008). Moreover, xc1

and Dc1 are the first cluster center and its density measure respectively. According to this equation, the density measure of the points adjacent to the first cluster center decreases dramatically and, as a result, the

probability of being selected as the next cluster center rapidly diminishes for these points. In this manner,

the algorithm repeatedly finds other cluster center using the above-mentioned equation.

Information Criteria Information Criteria (IC) have been successfully used in evaluation of data-driven models as alternative

of the laborious cross validation method (Terzaghi, 1943). IC do not split the data into different portions

and perform the training over the whole data set. In order to avoid excessive complexity of the model, a penalty function is defined which its value increases as the model becomes more complex. Generally, an

information criterion is a function of model’s parameters (n) as well as the number of training data (N).

As an example consider SRC criterion: SRC(n) = ln(MSE) +n.(ln(N)/N) (10)

where MSE is mean squared error over the whole data. If MSE is minimized as the only measure of

model’s performance, the problem of over-fitting will be inevitable especially when a few number of

training data is available. However, application of IC such as SRC results in an optimized trade-off between accuracy and complexity (Impe et al., 1997).




Research Article


Proposed Intelligent Models

Collecting Experimental Data

Stuedlein (Wang and Yen, 1999) collected results of load tests on stone columns and discussed that such data should fulfill a number of criteria in order to form a reliable database for statistical analysis. These

criteria mainly include: adequate representation of soil characteristics and load testing configuration,

uniformity of the soil (in failure zone), loading in a rapid manner and possibility of bearing capacity extrapolation from measured displacements. In current research, the method discussed by Studlein (Wang

and Yen, 1999) is adopted and results of 29 load tests over reinforced grounds are collected in Table 1.

This table presents bearing pressures measured at different settlements (2, 5, 10, 17, 25, 35 and 50 mm) as

well as the ultimate bearing capacity of composite ground. The parameters represent ground characteristics and configuration of the reinforcement system (i.e. stone columns) include: undrained

shear strength of surrounding soil (su), area replacement ratio (ar), width of the footing (B), depth of

footing embedment (df), diameter of stone column (dp) and its length (Lp).

Table 1: Experimental data collected from the literature to train and evaluate proposed intelligent

models

su

(kPa

)

ar

(%)

B

(m)

df

(m)

dp

(m)

Lp

(m)

Bearing pressure (kPa) for specific settlements

(mm)

reference 2 5 10 17 25 35 50

ultim

ate

30 100 0.3 0.0 0.3 8 133 244 358 462 541 605 660 722

Lillis et

al., (2004)

30 44.4 0.45

0.0 0.3 8 66 119 176 227 263 292 316 396

Lillis et

al.,

(2004)

30 25 0.6 0.0 0.3 8 50 79 109 143 177 216 268 559

Lillis et

al.,

(2004)

30 16 0.7

5 0.0 0.3 8 36 70 117 173 229 288 358 482

Lillis et al.,

(2004)

30 34.6 2.2

9

0.4

6

0.7

6

2.3

3 46 93 140 171 187 198 210 338

Stuedlein

, (2008)

30 34.6 2.29

0.46

0.76

4.64

51 102 153 191 218 244 276 477 Stuedlein, (2008)

30 100 0.7

6

0.4

6

0.7

6

2.3

3 154 292 398 455 485 510 539 604

Stuedlein

, (2008)

30 100 0.76

0.46

0.76

4.64

213 368 485 558 599 628 653 664 Stuedlein, (2008)

59 30.2 2.74

0.0 0.74

4.57

42 71 99 127 153 182 219 555

Wang

and Yen

(1999)

54 24.2 2.7

4 0.0

0.7

4

4.5

7 34 55 80 107 135 167 208 532

Wang and Yen

(1999)

59 30.2 2.7

4 0.0

0.7

4

3.0

5 50 96 147 199 246 297 358 645

Wang

and Yen (1999)




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75 30.2 2.74

0.0 0.76

4.57

45 94 149 196 234 272 319 624

Wang

and Yen

(1999)

65 30.2 2.7

4 0.0

0.7

4

4.5

7 48 95 145 194 238 282 335 615

Wang and Yen

(1999)

65 100 0.7

6

0.6

1

0.7

6

3.0

5 210 348 461 552 624 694 774 1096

Wang and Yen

(1999)

69 100 0.7

6

0.6

1

0.7

6

3.0

5 139 258 364 443 498 546 599 1006

Wang

and Yen (1999)

67 100 0.76

0.61

0.76

4.57

239 433 601 724 808 881 957 1132

Wang

and Yen

(1999)

70 100 0.7

6

0.6

1

0.7

6

4.5

7 163 281 377 449 502 552 612 1202

Wang and Yen

1999)

57 95 0.7

6

0.6

1

0.7

4

3.0

5 135 245 375 506 616 721 835 1115

Wang

and Yen (1999)

61 100 0.76

0.61

0.76

3.05

167 329 493 628 727 812 901 1093

Wang

and Yen

(1999)

63 88 0.7

6

0.6

1

0.7

1

3.0

5 173 310 463 607 721 819 916 1067

Wang

and Yen

(1999)

61 95 0.7

6

0.6

1

0.7

4

4.5

7 244 426 621 797 929 1038 1139 1214

Wang and Yen

(1999)

53 95 0.7

6

0.6

1

0.7

4

4.5

7 206 341 481 609 709 797 887 1071

Wang

and Yen (1999)

52 95 0.76

0.61

0.74

4.57

66 142 232 318 388 454 532 1106

Wang

and Yen

(1999)

12 46.8 1.3

7 0.0 1 5 102 146 173 183 185 185 186 189

White et al.,

(2007)

44 40.1 0.9

1

0.6

1

0.6

1 2.9 80 141 200 249 282 307 325 399

Wissman

n (1999)

12 36 1.2

5 0.0

0.8

5 14 34 51 65 78 90 101 114 177

Zadeh

(1965)

12 36 1.2

5 0.0

0.8

5 14 19 36 54 70 85 99 117 252

Zadeh

(1965)

12 100 0.85

0.0 0.85

14 94 130 156 179 199 218 241 378 Zadeh (1965)

100 100 0.6

1 0.0

0.6

1

3.0

5 43 116 236 390 542 694 860 1346 [34]




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Evaluating the Bearing Capacity

In this section, four intelligent models including two fuzzy models, ANFIS based on fuzzy c-means

clustering ANFIS (FCM) and subtractive clustering method ANFIS (SCM), and two neural network models, generalized regression neural network (GRNN) and feed-forward neural network (FNN), are

utilized to evaluate ultimate bearing capacity of spread footing on composite ground reinforced with stone

columns.

Figure 2: Experimental bearing capacity values versus predictions of proposed models

ANFIS

(FCM)

ANFIS

(SCM)




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All of the models have been developed using MATLAB software. With regard to Eq. 5 the first six

columns of Table 1 are selected as input parameters for the intelligent models. Furthermore, the

information criterion given by Eq. 10 is used in training and verification of the models. As stated earlier, there is no need to split the data into training and testing parts when an information criterion is applied.

Predictions of the four intelligent models are illustrated versus experimental bearing capacity in Figure 2.

Results of MLR model proposed by Stuedlein and Holtz (2013) are also shown in Figure 3. According to these two figures, all of the proposed intelligent models are able to precisely capture the value of ultimate

bearing capacity of reinforced grounds. In addition, GRNN and ANFIS (FCM) are of higher accuracy

compared with other two models.

Figure 3: Comparison of experimental bearing capacity values with predictions MLR model

proposed by Stuedlein and Holtz (2013)

Evaluating the Settlement

The modeling approach introduced in previous section is also used to estimate bearing pressures corresponding to specific settlements. Input parameters are those of the models for bearing capacity in

addition to the corresponding settlement. As a result, each model can evaluate the bearing pressure of

composite ground. Figure 4 depicts the outputs of models in comparison to experimental data. It is evident from this figure

that estimations of the intelligent models coincide well with experimental measurements. Predictions of

MLR model derived by Stuedlein and Holtz (2014) are illustrated in Figure 5 against real values of bearing pressure at different settlements. Comparison of this model with those illustrated in Figure 4

demonstrates high accuracy of the proposed intelligent models.

In addition, the settlement of spread footing over composite ground is evaluated in this section using

afore-mentioned models. The values of target function, i.e. the settlement, are interpolated from the data given in Table 1 in regard to a factor of safety equal to three (F.S. = 3). Figure 6 presents the outputs of

proposed models against interpolated settlements. It can be seen from this figure that the proposed models

can accurately evaluate the settlement.




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Figure 4: Experimental bearing pressures (corresponding to specific settlements) versus predictions

of proposed models

ANFIS

(FCM)

ANFIS

(SCM)




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Figure 5: Comparison of experimental bearing pressures with predictions MLR model proposed by

Stuedlein and Holtz (2014)

Figure 6: Interpolated settlements (with S.F=3) versus predictions of proposed models

ANFIS

(FCM)

ANFIS

(SCM)




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Concluding Remarks

In current study, four intelligent models were proposed to evaluate bearing capacity and settlement of

spread footing on clay ground reinforced by stone columns. The models include two ANFIS models (with c-means and subtractive clustering algorithms) and two neural networks (feed-forward and generalized

regression). Moreover, optimized design of models was achieved through an information criterion namely

SRC. Use of information criteria is particularly helpful when a small number of data, in regard to the model’s parameters, is available (as is the case in current study).

The data required for training and verification of the models were collected from high quality results

published in the literature. Due to application of information criterion, the database should not split into

separate parts. It was depicted that the four intelligent models are able to precisely estimate bearing capacity as well as bearing pressure of spread footing lying over clay ground improved by granular piles.

All of the models are of considerably higher accuracy compared with recently developed MLR models.

Furthermore, the models were able to precisely estimate the settlement with a safety factor equal to three. Among the proposed models, GRNN and ANFIS with c-means clustering showed the best performance in

all three modeling tasks.

ACKNOWLEDGEMENT

The authors are grateful to Seyed Abdolkhalegh Yadegarnejad for their valuable contributions.

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